Author’s abstract: We introduce an invariant of a \(K3\) surface with \(\mathbb Z_2\)-action equipped with a \(\mathbb Z_2\)-invariant Kähler metric, which we obtain using the equivariant analytic torsion of the trivial line bundle. This invariant is shown to be independent of the choice of the Kähler metric. It can be viewed as a function on the moduli space of \(K3\) surfaces with involution. The main result of this paper is that this function can be identified with an automorphic form, which characterizes the discriminant locus. In particular, we show that the Ray-Singer analytic torsion of the trivial line bundle on an Enriques surface with Ricci-flat Kähler metric is given by the value of the norm of the Borcherds \(\Phi\)-function and its period point.